稳定子映射的连续性与不可约扩张

Generic stabilizers in actions of simple algebraic groups II: higher Grassmannian varieties

This is the second of two papers treating faithful actions of simple algebraic groups on irreducible modules and on the associated Grassmannian varieties; in the first paper we considered the module itself and its projective space, while here we handle the varieties comprising the subspaces of the module of fixed dimension greater than one. By explicit calculation, we show that in each case, with essentially one exception, there is a dense open subset any point of which has stabilizer conjugate to a fixed subgroup, called the generic stabilizer. We provide tables listing generic stabilizers in…

简单代数群在正交与辛 Grassmannians 上作用的通用稳定子

Generic stabilizers for simple algebraic groups acting on orthogonal and symplectic Grassmannians

本文研究简单代数群在自对偶不可约模及全奇异子空间簇上的忠实作用，证明了除有限情况外存在通用稳定子，并据此解决了经典群中具有稠密双陪集的极大连通子群对的确定问题。

简单代数群作用中的一般稳定子

Generic stabilizers in actions of simple algebraic groups

本文通过显式计算，研究了简单代数群在不可约模及格拉斯曼流形上的忠实作用，证明了除特征2的一个例外外，一般稳定子存在，并提供了非平凡情形的列表及稠密或正则轨道的判定。

Generically free representations II: irreducible representations

We determine which faithful irreducible representations $V$ of a simple linear algebraic group $G$ are generically free for Lie($G$), i.e., which $V$ have an open subset consisting of vectors whose stabilizer in Lie($G$) is zero. This relies on bounds on $\dim V$ obtained in prior work (part I), which reduce the problem to a finite number of possibilities for $G$ and highest weights for $V$, but still infinitely many characteristics. The remaining cases are handled individually, some by computer calculation. These results were previously known for fields of characteristic zero, although new ph…

Rigidity of invariant convex sets in symmetric spaces

The main result implies that a proper convex subset of an irreducible higher rank symmetric space cannot have Zariski dense stabilizer.

IBIS soluble linear groups

Let $G$ be a finite permutation group on $Ω.$ An ordered sequence $(ω_1,\dots, ω_t)$ of elements of $Ω$ is an irredundant base for $G$ if the pointwise stabilizer is trivial and no point is fixed by the stabilizer of its predecessors. If all irredundant bases of $G$ have the same cardinality, $G$ is said to be an IBIS group. In this paper we give a classification of quasi-primitive soluble irreducible IBIS linear groups, and we also describe nilpotent and metacyclic IBIS linear groups and IBIS linear groups of odd order.

Construction of Fischer's sporadic group Fi'_{24} inside GL_{8671}(13)

In this article we construct an irreducible simple subgroup G = <q, y, t, w> of GL_{8671}(13) from an irreducible subgroup T of GL_{11}(2) isomorphic to Mathieu's simple group M_{24} by means of Algorithm 2.5 of [13]. We also use the first author's similar construction of Fischer's sporadic simple group G_1 = Fi_{23} described in [11]. He starts from an irreducible subgroup T_1 of GL_{11}(2) contained in T which is isomorphic to M_{23}. In [7] J. Hall and L. S. Soicher published a nice presentation of Fischer's original 3-transposition group Fi_{24} [6]. It is used here to show that G is isomo…

特征的$p$-有理性的连续性与主块

The continuity of $p$-rationality of characters and the principal block

本文研究有限群不可约特征的有理性，证明$2$-有理性连续性可在主$2$-块中探测以改进前人结果，并为所有素数提出$1$级连续性间隙的群论准则猜想。

The continuity of $p$-rationality and a lower bound for $p'$-degree characters of finite groups

Let $p$ be a prime and $G$ a finite group. We propose a strong bound for the number of $p'$-degree irreducible characters of $G$ in terms of the commutator factor group of a Sylow $p$-subgroup of $G$. The bound arises from a recent conjecture of Navarro and Tiep [NT21] on fields of character values and a phenomenon called the continuity of $p$-rationality level of $p'$-degree characters. This continuity property in turn is predicted by the celebrated McKay-Navarro conjecture [Nav04]. We achieve both the bound and the continuity property for $p=2$.

Moduli Spaces of Unordered $n\ge5$ Points on the Riemann Sphere and Their Singularities

For $n\ge5$, it is well known that the moduli space $\mathfrak{M_{0,\:n}}$ of unordered $n$ points on the Riemann sphere is a quotient space of the Zariski open set $K_n$ of $\mathbb C^{n-3}$ by an $S_n$ action. The stabilizers of this $S_n$ action at certain points of this Zariski open set $K_n$ correspond to the groups fixing the sets of $n$ points on the Riemann sphere. Let $α$ be a subset of $n$ distinct points on the Riemann sphere. We call the group of all linear fractional transformations leaving $α$ invariant the stabilizer of $α$, which is finite by observation. For each non-trivial f…

The stable representations of $\mathrm{GL}_{N}$ over finite local principal ideal rings

Let $\mathcal{O}$ be a discrete valuation ring with maximal ideal $\mathfrak{p}$ and with finite residue field $\mathbb{F}_{q}$, the field with $q$ elements where $q$ is a power of a prime $p$. For $r \ge 1$, we write $\mathcal{O}_r$ for the reduction of $\mathcal{O}$ modulo the ideal $\mathfrak{p}^r$. An irreducible ordinary representation of the finite group $\mathrm{GL}_{N}(\mathcal{O}_{r})$ is called stable if its restriction to the principal congruence kernel $K^l=1+\mathfrak{p}^{l}\mathrm{M}_{N}(\mathcal{O}_r)$, where $l=\lceil \frac{r}{2} \rceil$, consists of irreducible representations…

Modules for algebraic groups with finitely many orbits on totally singular 2-spaces

This is the author's second paper treating the double coset problem for classical groups. Let $G$ be an algebraic group over an algebraically closed field $K$. The double coset problem consists of classifying the pairs $H,J$ of closed connected subgroups of $G$ with finitely many $(H,J)$-double cosets in $G$. The critical setup occurs when one of $H,J$, say $H$, is reductive, and $J$ is a parabolic subgroup. Assume that $G$ is a classical group, $H$ is simple and $J$ is a maximal parabolic $P_k$, the stabilizer of a totally singular $k$-space. Then most candidates have $k=1$ or $k=2$. The case…

通过群傅里叶分解刻画量子资源性

Characterizing quantum resourcefulness via group-Fourier decompositions

本文提出基于群傅里叶分解的通用框架，通过GFD纯度刻画量子态资源性，发现低资源态位于低维不可约表示，高资源态位于高维，并具有操作意义。

轨道异常、各向同性不连续性和大同构分量

Orbit misbehavior, isotropy discontinuity, and large isotypic components

本文研究了紧群作用下，同构分量作为不变子代数模的有限生成性条件，以及紧群作用在最大同调紧缩上的管状结构与群的李群性质的关系。